Remark  (Numerical behavior of Algorithm  and Algorithm REF ) 
As noted in Section , lines 1-14 of Algorithm , i.e., the exact algorithm, are based on an implicit LU decomposition of the tridiagonal matrix \(T_j\)  as in (), i.e., a coupled two-term recurrence. The exact algorithm does not require the storage of the matrix \(T_j\) . Additionally, the solutions to the subsequent shifted systems in (REF ) are computed without solving a linear system analogous to (REF ). In order to fit the flexible framework as introduced in [1]} and allow for the solution to many shifted system simultaneously, Algorithm REF , i.e., the inexact algorithm, computes and stores the matrix \(T_j\)  and performs 1 linear solve for each \(\mu \)  of interest. In this way, the residual vectors in the inexact algorithm are updated via a three-term recurrence.
